Let $G$ be a finite group such that it has a normal Sylow p-subgroup. Is there any non-trivial element in the center of $G$?
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@DonAntonio: Thanks. The question edited. – Tim Apr 09 '13 at 10:20
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1Relevant. – anon Apr 09 '13 at 10:21
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Well, there is always the unity element in the center, so I gather you meant "non-trivial" element, and counterexample: $\,S_3\,$ ...
What's its normal Sylow subgroup and why $\,Z(S_3)=1\,$ ?
DonAntonio
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For $p = 2$, the alternating group $A_4$ should be a counterexample. For $p > 2$ you could try the dihedral group $D_{2p}$. There are nonsolvable examples for $p > 2$ (see the question linked in the comments), but none for $p = 2$ by Feit-Thompson.
Mikko Korhonen
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It is worth pointing out that $S_3\cong D_6$, the dihedral group of order $6$. So this answer is, in some ways, a natural and rather neat extension of DonAntonio's answer. – user1729 Apr 09 '13 at 15:20